where \(f_q\)  denotes the Littlewood-Paley projection onto the \(q\) th dyadic shell in Fourier space, see [1]}. For any \(q\in \mathbb {N}\)  we also denote
\(f_{<q} = \sum _{p<q} f_p, \qquad f_{\sim q} = \sum _{q-2\le p\le q+2} f_p.\) 
